Skip to main content

Full text of "Analytical Mechanics"

See other formats

177.   Newtonian Potential.  The potential energy of a unit mass placed at a point of a Newtonian field is called the potential at that point.    The standard configuration or the position of zero potential is taken to be infinitely far from the center of the field.    But the potential energy of a body equals the work done in bringing the body from the position of zero potential energy, therefore the following definition is equivalent to the one just given.
The potential at a point equals the work done in bringing a unit mass from an infinite distance to that point.
178.   Potential Due to a Single Particle.  Let m be the mass of the particle, U the potential energy of a particle of mass m' placed in the field of force of the -first particle, r the distance between the two particles, and V the potential at the position of mr due to m.    Then by the definitions of V and U
7 =
where F is the force experienced by m! due to the field of m.
T>   x                                   n              mm/
But                            F =  7  
Therefore                   7= ym I   
J co   r~
= -Tf-                               (VI)
The negative sign indicates the fact that when a particle is brought to the field of another attracting particle work will be done by the particle and not by the agent which brings it. Therefore the potential due to a material particle, as we have defined it, is everywhere negative, except at infinity where it is zero. In case of electrical and magnetic masses potential is defined as the work done in bringing a unit posi* tive charge, or unit positive pole, from infinity. Therefore